Question: Given $ \overrightarrow{OA}\perp\overrightarrow{OC}$, $ m \angle AOB = 2x + 49$, and $ m \angle BOC = 4x + 11$, find $m\angle BOC$. $O$ $A$ $C$ $B$
Answer: From the diagram, we see that together ${\angle AOB}$ and ${\angle BOC}$ form ${\angle AOC}$ , so $ {m\angle AOB} + {m\angle BOC} = {m\angle AOC}$ Since we are given that $\overrightarrow{OA}\perp\overrightarrow{OC}$ , we know ${m\angle AOC = 90}$ Substitute in the expressions that were given for each measure: $ {2x + 49} + {4x + 11} = {90}$ Combine like terms: $ 6x + 60 = 90$ Subtract $60$ from both sides: $ 6x = 30$ Divide both sides by $6$ to find $x$ $ x = 5$ Substitute $5$ for $x$ in the expression that was given for $m\angle BOC$ $ m\angle BOC = 4({5}) + 11$ Simplify: $ {m\angle BOC = 20 + 11}$ So ${m\angle BOC = 31}$.